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Properties

Label 968.2.i.o
Level $968$
Weight $2$
Character orbit 968.i
Analytic conductor $7.730$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [968,2,Mod(9,968)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(968, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 968.i (of order \(5\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.72951891566\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.324000000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{2}) q^{3} + ( - 2 \beta_{6} - \beta_1) q^{5} + (\beta_{6} + \beta_{4} - \beta_{3} + \cdots + 1) q^{7}+ \cdots + ( - 2 \beta_{7} - 2 \beta_{5} + \cdots - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{2}) q^{3} + ( - 2 \beta_{6} - \beta_1) q^{5} + (\beta_{6} + \beta_{4} - \beta_{3} + \cdots + 1) q^{7}+ \cdots + (4 \beta_{7} + 4 \beta_{5} + \cdots + 4 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 4 q^{5} + 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 4 q^{5} + 2 q^{7} - 2 q^{9} + 2 q^{13} + 10 q^{15} + 8 q^{17} + 10 q^{19} - 32 q^{21} + 24 q^{23} - 4 q^{25} - 8 q^{27} - 2 q^{29} + 6 q^{31} - 10 q^{35} - 8 q^{37} + 14 q^{39} + 12 q^{41} - 64 q^{43} - 64 q^{45} + 10 q^{47} + 6 q^{49} + 2 q^{51} + 8 q^{53} + 4 q^{57} - 20 q^{59} + 20 q^{61} + 14 q^{63} + 64 q^{65} + 40 q^{67} + 12 q^{69} - 12 q^{71} - 4 q^{73} - 28 q^{75} - 2 q^{79} - 2 q^{81} - 6 q^{83} - 10 q^{85} + 56 q^{87} - 104 q^{89} - 14 q^{91} + 12 q^{93} - 14 q^{95} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 27 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27\beta_{7} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(727\) \(849\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{2} - \beta_{4} - \beta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
0.535233 + 1.64728i
−0.535233 − 1.64728i
1.40126 − 1.01807i
−1.40126 + 1.01807i
1.40126 + 1.01807i
−1.40126 − 1.01807i
0.535233 − 1.64728i
−0.535233 + 1.64728i
0 −2.21028 + 1.60586i 0 −1.15327 − 3.54939i 0 2.21028 + 1.60586i 0 1.37948 − 4.24561i 0
9.2 0 0.592242 − 0.430289i 0 −0.0828009 − 0.254835i 0 −0.592242 − 0.430289i 0 −0.761449 + 2.34350i 0
81.1 0 −0.226216 + 0.696222i 0 0.216775 − 0.157497i 0 0.226216 + 0.696222i 0 1.99350 + 1.44836i 0
81.2 0 0.844250 − 2.59833i 0 3.01929 − 2.19364i 0 −0.844250 − 2.59833i 0 −3.61153 − 2.62393i 0
729.1 0 −0.226216 − 0.696222i 0 0.216775 + 0.157497i 0 0.226216 − 0.696222i 0 1.99350 − 1.44836i 0
729.2 0 0.844250 + 2.59833i 0 3.01929 + 2.19364i 0 −0.844250 + 2.59833i 0 −3.61153 + 2.62393i 0
753.1 0 −2.21028 − 1.60586i 0 −1.15327 + 3.54939i 0 2.21028 − 1.60586i 0 1.37948 + 4.24561i 0
753.2 0 0.592242 + 0.430289i 0 −0.0828009 + 0.254835i 0 −0.592242 + 0.430289i 0 −0.761449 − 2.34350i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.2.i.o 8
11.b odd 2 1 968.2.i.n 8
11.c even 5 1 968.2.a.k ✓ 2
11.c even 5 3 inner 968.2.i.o 8
11.d odd 10 1 968.2.a.l yes 2
11.d odd 10 3 968.2.i.n 8
33.f even 10 1 8712.2.a.bs 2
33.h odd 10 1 8712.2.a.br 2
44.g even 10 1 1936.2.a.p 2
44.h odd 10 1 1936.2.a.q 2
88.k even 10 1 7744.2.a.cw 2
88.l odd 10 1 7744.2.a.cx 2
88.o even 10 1 7744.2.a.bu 2
88.p odd 10 1 7744.2.a.bv 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
968.2.a.k ✓ 2 11.c even 5 1
968.2.a.l yes 2 11.d odd 10 1
968.2.i.n 8 11.b odd 2 1
968.2.i.n 8 11.d odd 10 3
968.2.i.o 8 1.a even 1 1 trivial
968.2.i.o 8 11.c even 5 3 inner
1936.2.a.p 2 44.g even 10 1
1936.2.a.q 2 44.h odd 10 1
7744.2.a.bu 2 88.o even 10 1
7744.2.a.bv 2 88.p odd 10 1
7744.2.a.cw 2 88.k even 10 1
7744.2.a.cx 2 88.l odd 10 1
8712.2.a.br 2 33.h odd 10 1
8712.2.a.bs 2 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):

\( T_{3}^{8} + 2T_{3}^{7} + 6T_{3}^{6} + 16T_{3}^{5} + 44T_{3}^{4} - 32T_{3}^{3} + 24T_{3}^{2} - 16T_{3} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} - 2T_{7}^{7} + 6T_{7}^{6} - 16T_{7}^{5} + 44T_{7}^{4} + 32T_{7}^{3} + 24T_{7}^{2} + 16T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$17$ \( T^{8} - 8 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{7} + \cdots + 234256 \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T - 18)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} + 2 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$31$ \( T^{8} - 6 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$41$ \( T^{8} - 12 T^{7} + \cdots + 1185921 \) Copy content Toggle raw display
$43$ \( (T + 8)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} + \cdots + 234256 \) Copy content Toggle raw display
$53$ \( T^{8} - 8 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$59$ \( T^{8} + 20 T^{7} + \cdots + 59969536 \) Copy content Toggle raw display
$61$ \( T^{8} - 20 T^{7} + \cdots + 59969536 \) Copy content Toggle raw display
$67$ \( (T^{2} - 10 T - 2)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} + 12 T^{7} + \cdots + 331776 \) Copy content Toggle raw display
$73$ \( T^{8} + 4 T^{7} + \cdots + 3748096 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 3429742096 \) Copy content Toggle raw display
$83$ \( T^{8} + 6 T^{7} + \cdots + 104976 \) Copy content Toggle raw display
$89$ \( (T^{2} + 26 T + 157)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} - 10 T^{7} + \cdots + 279841 \) Copy content Toggle raw display
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